Accurate and efficient simulations of Hamiltonian mechanical systems with discontinuous potentials

This article considers Hamiltonian mechanical systems with potential functions admitting jump discontinuities. The focus is on accurate and efficient numerical approximations of their solutions, which will be defined via the laws of reflection and refraction. Despite of the success of symplectic integrators for smooth mechanical systems, their construction for the discontinuous ones is nontrivial, and numerical convergence order can be impaired too. Several rather-usable numerical methods are proposed, including: a first-order symplectic integrator for general problems, a third-order symplectic integrator for problems with only one linear interface, arbitrarily high-order reversible integrators for general problems (no longer symplectic), and an adaptive time-stepping version of the previous high-order method. Interestingly, whether symplecticity leads to favorable long time performance is no longer clear due to discontinuity, as traditional Hamiltonian backward error analysis does not apply any more. Therefore, at this stage, our recommended default method is the last one. Various numerical evidence, on the order of convergence, long time performance, momentum map conservation, and consistency with the computationally-expensive penalty method, are supplied. A complex problem, namely the Sauteed Mushroom, is also proposed and numerically investigated, for which multiple bifurcations between trapped and ergodic dynamics are observed. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This article focuses on accurate and efficient numerical approximations of solutions for Hamiltonian mechanical systems with potential functions admitting jump discontinuities. It proposes several numerical methods and provides numerical evidence on their convergence, performance, and consistency.